1.
Stat310 Continuous variables
Hadley Wickham
Tuesday, 3 February 2009
2.
1. Notes about the exam
2. Finish off Poisson
3. Introduction to continuous variables
4. The uniform distribution
Tuesday, 3 February 2009
3.
Exam
• Exam structure
• Grading tomorrow
• Purpose of notes
• Question 1 - most of you managed to get
it (eventually) - at least 3 different ways
• Question 2 & 4 - did really well
• Question 3 - more of a struggle
Tuesday, 3 February 2009
4.
Poisson distribution
X = Number of times some event happens
If number of events occurring in non-
overlapping times is independent, and
Probability of exactly one event occurring
in short interval of length h is ∝ λh, and
Probability of two or more events in a
sufﬁciently short internal is basically 0
Then X ~ Poisson(λ)
Tuesday, 3 February 2009
5.
Examples
Number of calls to a switchboard
Number of eruptions of a volcano
Number of alpha particles emitted from a
radioactive source
Number of defects in a roll of paper
Tuesday, 3 February 2009
7.
What is λ?
• What is the sample space of X?
• Let’s start by looking at the mean and
variance of X.
• How?
Tuesday, 3 February 2009
8.
What is λ?
• λ is the mean rate of events per unit
time.
• If you change the unit of time from 1 to
t, you’ll expect λt events - another
Poisson process/distribution
• ie. if X ~ Poisson(λ), and Y = tX, then Y
~ Poisson(λt)
Tuesday, 3 February 2009
9.
Example
• A small amount of radioactive material
emits one alpha particle on average
every second. If we assume it is a
Poisson process, then:
• How many particles would be emitted
ever minute, on average?
• What is the probability that no particles
are emitted in 10 seconds?
Tuesday, 3 February 2009
10.
Continuous random
variables
Tuesday, 3 February 2009
11.
Continuous r.v.
• Sample space is the real line
• Mathematical tools: more differentiation
+ integration
• Same vocabulary, slightly different
deﬁnitions
• New distributions
Tuesday, 3 February 2009
12.
Intuition
Imagine you have a spinner which is
equally likely to point in any direction. Let
X be the angle the spinner points.
What is P(X ∈ [0, 90]) ? What is P(X ∈
[270, 90]) ? What is P(X ∈ [70, 98]) ?
What is the general formula?
What is P(X = 90) ?
Tuesday, 3 February 2009
13.
Cumulative distribution function
x
F (x) = P (X ≤ x) = f (t)dt
−∞
b
P (X ∈ [a, b]) = f (x)dx = F (b) − F (a)
a
P (X = a) = P (x = [a, a]) = F (a) − F (a) = 0
Tuesday, 3 February 2009
14.
f (x) For continuous x,
f(x) is a probability
density function.
Not a probability!
Tuesday, 3 February 2009
15.
f (x)
Integrate Differentiate
F(x)
Tuesday, 3 February 2009
16.
Conditions
f (x) ≥ 0 ∀x ∈ R
f (x) = 1
R
Tuesday, 3 February 2009
17.
Questions?
Is f(x) < 1 for all x?
What do those conditions imply about
F(x)?
Tuesday, 3 February 2009
18.
E(u(X)) = u(x)f (x)dx
R
MX (t) = e f (x)dx
tx
R
Tuesday, 3 February 2009
19.
The discrete uniform
Assigns probability uniformly in an interval
[a, b] of the real line
X ~ Uniform(a, b)
What are F(x) and f(x) ?
b
f (x)dx = 1
a
Tuesday, 3 February 2009
20.
Intuition
X ~ Unif(1, b)
What do you expect the mean of X to be?
What about the variance?
Tuesday, 3 February 2009
21.
a+b
E(X) =
2
(b − a)
2
V ar(X) =
12
Tuesday, 3 February 2009
22.
Question
X ~ Unif(0, 1)
Y = 10 X
What is the distribution of Y?
How does the variance of Y compare to
the variance of X?
Tuesday, 3 February 2009
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